Question

1. expand and simplify.

a) 4(x - 6)(x + 7)
b) -(x + 3)(4x - 1)
c) 6x(x + 1)²
d) (x + 4)(x - 2) + (x - 1)(x + 5)
e) (4x - 1)(4x + 1) - (x + 3)²
f) 2(3x + 4)² - 3(x - 2)²

Explanation:

Let's solve each part one by one:

Part a) \( 4(x - 6)(x + 7) \)

Step 1: Expand \( (x - 6)(x + 7) \) using the distributive property (FOIL method)

\( (x - 6)(x + 7)=x\times x+x\times 7-6\times x - 6\times7=x^{2}+7x-6x - 42=x^{2}+x - 42 \)

Step 2: Multiply the result by 4

\( 4(x^{2}+x - 42)=4\times x^{2}+4\times x-4\times42 = 4x^{2}+4x-168 \)

Part b) \( -(x + 3)(4x - 1) \)

Step 1: Expand \( (x + 3)(4x - 1) \) using the distributive property

\( (x + 3)(4x - 1)=x\times4x+x\times(- 1)+3\times4x+3\times(-1)=4x^{2}-x + 12x-3=4x^{2}+11x - 3 \)

Step 2: Multiply the result by - 1

\( - (4x^{2}+11x - 3)=-4x^{2}-11x + 3 \)

Part c) \( 6x(x + 1)^{2} \)

Step 1: Expand \( (x + 1)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a=x \) and \( b = 1 \)

\( (x + 1)^{2}=x^{2}+2x\times1+1^{2}=x^{2}+2x + 1 \)

Step 2: Multiply the result by \( 6x \)

\( 6x(x^{2}+2x + 1)=6x\times x^{2}+6x\times2x+6x\times1=6x^{3}+12x^{2}+6x \)

Part d) \( (x + 4)(x - 2)+(x - 1)(x + 5) \)

Step 1: Expand \( (x + 4)(x - 2) \)

\( (x + 4)(x - 2)=x\times x+x\times(-2)+4\times x+4\times(-2)=x^{2}-2x + 4x-8=x^{2}+2x - 8 \)

Step 2: Expand \( (x - 1)(x + 5) \)

\( (x - 1)(x + 5)=x\times x+x\times5-1\times x-1\times5=x^{2}+5x - x-5=x^{2}+4x - 5 \)

Step 3: Add the two expanded expressions

\( (x^{2}+2x - 8)+(x^{2}+4x - 5)=x^{2}+x^{2}+2x + 4x-8 - 5=2x^{2}+6x - 13 \)

Part e) \( (4x - 1)(4x + 1)-(x + 3)^{2} \)

Step 1: Expand \( (4x - 1)(4x + 1) \) using the difference of squares formula \( (a - b)(a + b)=a^{2}-b^{2} \), where \( a = 4x \) and \( b=1 \)

\( (4x - 1)(4x + 1)=(4x)^{2}-1^{2}=16x^{2}-1 \)

Step 2: Expand \( (x + 3)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a=x \) and \( b = 3 \)

\( (x + 3)^{2}=x^{2}+6x + 9 \)

Step 3: Subtract the second expanded expression from the first

\( (16x^{2}-1)-(x^{2}+6x + 9)=16x^{2}-1-x^{2}-6x - 9=15x^{2}-6x - 10 \)

Part f) \( 2(3x + 4)^{2}-3(x - 2)^{2} \)

Step 1: Expand \( (3x + 4)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a = 3x \) and \( b = 4 \)

\( (3x + 4)^{2}=(3x)^{2}+2\times3x\times4+4^{2}=9x^{2}+24x + 16 \)
Multiply by 2: \( 2(9x^{2}+24x + 16)=18x^{2}+48x + 32 \)

Step 2: Expand \( (x - 2)^{2} \) using the formula \( (a - b)^{2}=a^{2}-2ab + b^{2} \), where \( a=x \) and \( b = 2 \)

\( (x - 2)^{2}=x^{2}-4x + 4 \)
Multiply by - 3: \( - 3(x^{2}-4x + 4)=-3x^{2}+12x - 12 \)

Step 3: Subtract the two expanded expressions

\( (18x^{2}+48x + 32)+(-3x^{2}+12x - 12)=18x^{2}-3x^{2}+48x + 12x+32 - 12=15x^{2}+60x + 20 \)

Final Answers:

a) \( \boldsymbol{4x^{2}+4x - 168} \)

b) \( \boldsymbol{-4x^{2}-11x + 3} \)

c) \( \boldsymbol{6x^{3}+12x^{2}+6x} \)

d) \( \boldsymbol{2x^{2}+6x - 13} \)

e) \( \boldsymbol{15x^{2}-6x - 10} \)

f) \( \boldsymbol{15x^{2}+60x + 20} \)

Answer:

Let's solve each part one by one:

Part a) \( 4(x - 6)(x + 7) \)

Step 1: Expand \( (x - 6)(x + 7) \) using the distributive property (FOIL method)

\( (x - 6)(x + 7)=x\times x+x\times 7-6\times x - 6\times7=x^{2}+7x-6x - 42=x^{2}+x - 42 \)

Step 2: Multiply the result by 4

\( 4(x^{2}+x - 42)=4\times x^{2}+4\times x-4\times42 = 4x^{2}+4x-168 \)

Part b) \( -(x + 3)(4x - 1) \)

Step 1: Expand \( (x + 3)(4x - 1) \) using the distributive property

\( (x + 3)(4x - 1)=x\times4x+x\times(- 1)+3\times4x+3\times(-1)=4x^{2}-x + 12x-3=4x^{2}+11x - 3 \)

Step 2: Multiply the result by - 1

\( - (4x^{2}+11x - 3)=-4x^{2}-11x + 3 \)

Part c) \( 6x(x + 1)^{2} \)

Step 1: Expand \( (x + 1)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a=x \) and \( b = 1 \)

\( (x + 1)^{2}=x^{2}+2x\times1+1^{2}=x^{2}+2x + 1 \)

Step 2: Multiply the result by \( 6x \)

\( 6x(x^{2}+2x + 1)=6x\times x^{2}+6x\times2x+6x\times1=6x^{3}+12x^{2}+6x \)

Part d) \( (x + 4)(x - 2)+(x - 1)(x + 5) \)

Step 1: Expand \( (x + 4)(x - 2) \)

\( (x + 4)(x - 2)=x\times x+x\times(-2)+4\times x+4\times(-2)=x^{2}-2x + 4x-8=x^{2}+2x - 8 \)

Step 2: Expand \( (x - 1)(x + 5) \)

\( (x - 1)(x + 5)=x\times x+x\times5-1\times x-1\times5=x^{2}+5x - x-5=x^{2}+4x - 5 \)

Step 3: Add the two expanded expressions

\( (x^{2}+2x - 8)+(x^{2}+4x - 5)=x^{2}+x^{2}+2x + 4x-8 - 5=2x^{2}+6x - 13 \)

Part e) \( (4x - 1)(4x + 1)-(x + 3)^{2} \)

Step 1: Expand \( (4x - 1)(4x + 1) \) using the difference of squares formula \( (a - b)(a + b)=a^{2}-b^{2} \), where \( a = 4x \) and \( b=1 \)

\( (4x - 1)(4x + 1)=(4x)^{2}-1^{2}=16x^{2}-1 \)

Step 2: Expand \( (x + 3)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a=x \) and \( b = 3 \)

\( (x + 3)^{2}=x^{2}+6x + 9 \)

Step 3: Subtract the second expanded expression from the first

\( (16x^{2}-1)-(x^{2}+6x + 9)=16x^{2}-1-x^{2}-6x - 9=15x^{2}-6x - 10 \)

Part f) \( 2(3x + 4)^{2}-3(x - 2)^{2} \)

Step 1: Expand \( (3x + 4)^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a = 3x \) and \( b = 4 \)

\( (3x + 4)^{2}=(3x)^{2}+2\times3x\times4+4^{2}=9x^{2}+24x + 16 \)
Multiply by 2: \( 2(9x^{2}+24x + 16)=18x^{2}+48x + 32 \)

Step 2: Expand \( (x - 2)^{2} \) using the formula \( (a - b)^{2}=a^{2}-2ab + b^{2} \), where \( a=x \) and \( b = 2 \)

\( (x - 2)^{2}=x^{2}-4x + 4 \)
Multiply by - 3: \( - 3(x^{2}-4x + 4)=-3x^{2}+12x - 12 \)

Step 3: Subtract the two expanded expressions

\( (18x^{2}+48x + 32)+(-3x^{2}+12x - 12)=18x^{2}-3x^{2}+48x + 12x+32 - 12=15x^{2}+60x + 20 \)

Final Answers:

a) \( \boldsymbol{4x^{2}+4x - 168} \)

b) \( \boldsymbol{-4x^{2}-11x + 3} \)

c) \( \boldsymbol{6x^{3}+12x^{2}+6x} \)

d) \( \boldsymbol{2x^{2}+6x - 13} \)

e) \( \boldsymbol{15x^{2}-6x - 10} \)

f) \( \boldsymbol{15x^{2}+60x + 20} \)